UVALDE, Texas — Frustrated onlookers urged police officers to charge into the Texas elementary school where a gunman’s rampage killed 19 children and two teachers, witnesses said Wednesday, as investigators worked to track the massacre that lasted upwards of 40 minutes and ended when the 18-year-old shooter was killed by a Border Patrol team.

“Go in there! Go in there!” nearby women shouted at the officers soon after the attack began, Juan Carranza, 24, said. He saw the scene from outside his house, across the street from Robb Elementary School in the close-knit town of Uvalde. Mr. Carranza said the officers did not go in.

Javier Cazares, whose fourth grade daughter, Jacklyn Cazares, was killed in the attack, said he raced to the school when he heard about the shooting, arriving while police were still gathered outside the building.

Upset that police were not moving in, he raised the idea of charging into the school with several other bystanders.

“Let’s just rush in because the cops aren’t doing anything like they are supposed to,” he said. “More could have been done.”

“They were unprepared,” he added.

Minutes earlier, Mr. Carranza had watched as Salvador Ramos crashed his truck into a ditch outside the school, grabbed his AR-15-style semi-automatic rifle, and shot at two people outside a nearby funeral home who ran away uninjured.

Officials say he “encountered” a school district security officer outside the school, though there were conflicting reports from authorities on whether the men exchanged gunfire. After running inside, he fired on two arriving Uvalde police officers who were outside the building, the Texas Department of Public Safety spokeswoman, Travis Considine, said. The police officers were injured.

After entering the school, Ramos charged into one classroom and began to kill.

He “barricaded himself by locking the door and just started shooting children and teachers that were inside that classroom,” Lieutenant Christopher Olivarez of the Department of Public Safety told CNN. “It just shows you the complete evil of the shooter.”

All those killed were in the same classroom, he said.

The Department of Public Safety director, Steve McCraw, told reporters that 40 minutes to an hour elapsed from when Ramos opened fire on the school security officer to when the tactical team shot him, though a department spokesman said later that they could not give a solid estimate of how long the gunman was in the school or when he was killed.

“The bottom line is law enforcement was there,” Mr. McCraw said. “They did engage immediately. They did contain [Ramos] in the classroom.”

Meanwhile, a law enforcement official familiar with the investigation said the Border Patrol agents had trouble breaching the classroom door and had to get a staff member to open the room with a key. The official spoke on the condition of anonymity because he was not authorized to speak publicly about the ongoing investigation.

Mr. Carranza said the officers should have entered the school sooner.

“There were more of them. There was just one of him,” he said.

Uvalde is a largely Latino town of some 16,000 people about 75 miles from the Mexican border. Robb Elementary, which has nearly 600 students in second, third and fourth grades, is a single-story brick structure in a mostly residential neighborhood of modest homes.

Before attacking the school, Ramos shot and wounded his grandmother at the home they shared, authorities said.

Neighbor Gilbert Gallegos, 82, who lives across the street and has known the family for decades, said he was puttering in his yard when he heard the shots.

Ramos ran out the front door and across the small yard to the truck parked in front of the house. He seemed panicked, Mr. Gallegos said, and had trouble getting the truck out of park.

Then he raced away: “He spun out, I mean fast,” spraying gravel in the air.

His grandmother emerged covered in blood: “She says, ‘Berto, this is what he did. He shot me.’” She was hospitalized.

Mr. Gallegos, whose wife called 911, said he had heard no arguments before or after the shots, and knew of no history of bullying or abuse of Ramos, who he rarely saw.

Investigators also shed no light on Ramos’ motive for the attack, which also left at least 17 people wounded. Governor Abbott said Ramos, a resident of the small town about 85 miles west of San Antonio, had no known criminal or mental health history.

“We don’t see a motive or catalyst right now,” Mr. McCraw of the Department of Public Safety said.

Ramos legally bought the rifle and a second one like it last week, just after his birthday, authorities said.

About a half-hour before the mass shooting, Ramos sent the first of three online messages warning about his plans, Mr. Abbott said.

Ramos wrote that he was going to shoot his grandmother, then that he had shot the woman. In the last note, sent about 15 minutes before he reached Robb Elementary, he said he was going to shoot up an elementary school, according to Mr. Abbott. Investigators said Ramos did not specify which school.

Ramos sent the private, one-to-one text messages via Facebook, a company spokesman, Andy Stone, said. It was not clear who received the messages.

Amid calls for tighter restrictions on firearms, the Republican governor repeatedly talked about mental health struggles among Texas young people and argued that tougher gun laws in Chicago, New York, and California are ineffective.

Representative Beto O’Rourke, who is running against Mr. Abbott for governor, interrupted Wednesday’s news conference, calling the tragedy “predictable.” Pointing his finger at Mr. Abbott, he said: “This is on you until you choose to do something different. This will continue to happen.” 

Mr. O’Rourke was escorted out as some in the room yelled at him. 

The attack was the deadliest school shooting in the U.S. since a gunman killed 20 children and six adults at Sandy Hook Elementary in Newtown, Connecticut, in December 2012.